Introduction to Transformations

A transformation moves every point of a figure according to a rule. The original figure is called the preimage, and the new figure is called the image. We mark image points with a prime symbol: the image of point $A$ is $A'$ , read " $A$ prime." Matching up preimage points with their primed images is how you figure out what a transformation did.

There are four transformations to know: translations slide, reflections flip, rotations turn, and dilations resize. Three of the four leave the figure exactly the same size and shape — those are the rigid motions — and one changes the size. Sorting the four types and knowing which are rigid is the whole job of this lesson.

The four transformations

A translation slides every point of the figure the same distance in the same direction. Nothing turns, nothing flips — the image is the same figure in a new spot.

A reflection flips the figure across a line, called the line of reflection. The image is a mirror copy: same size, but facing the other way.

A rotation turns the figure about a fixed point, called the center of rotation, through some number of degrees.

A dilation resizes the figure by a scale factor, measured from a center point. A scale factor greater than $1$ enlarges the figure; a scale factor between $0$ and $1$ shrinks it.

Rigid motions

A rigid motion is a transformation whose image is congruent to its preimage — the same size and the same shape, with every side length and every angle measure preserved. Translations, reflections, and rotations are all rigid motions. Sliding, flipping, or turning a figure moves it without distorting it.

A dilation is not a rigid motion, because it changes the size of the figure. The image of a dilation is similar to the preimage — same shape, matching angles — but not congruent unless the scale factor happens to be exactly $1$ .

This gives you a fast test on any transformation question: if the image is a different size than the preimage, the transformation was a dilation. If it is the same size, it was a slide, flip, or turn.

Reading a transformation from coordinates

On the coordinate plane, compare each preimage point with its image. If every point moved by the same amounts — say, $4$ left and $5$ up — the rule is a translation: $(x, y) \to (x - 4, y + 5)$ .

If every image point has the opposite $x$ -coordinate and the same $y$ -coordinate, the rule $(x, y) \to (-x, y)$ is a reflection across the $y$ -axis. Opposite $y$ with the same $x$ , the rule $(x, y) \to (x, -y)$ , is a reflection across the $x$ -axis.

Always check more than one vertex. A single matching point can fit several different transformations; two or three matching points pin the rule down.

Worked examples

Example 1: translate a point

Point $P(2, 5)$ is translated $3$ units left and $4$ units down. Find the coordinates of $P'$ .

Left

3

subtracts from the

x

-coordinate

2 - 3 = -1

Down

4

subtracts from the

y

-coordinate

5 - 4 = 1

Write the image point

P' = (-1, 1)

Answer: $P' = (-1, 1)$

Example 2: name the transformation from a rule

A transformation maps $(3, 2) \to (3, -2)$ and $(1, 5) \to (1, -5)$ . What kind of transformation is it?

Compare coordinates: each

x

stays the same

3 \to 3, \quad 1 \to 1

Each

y

becomes its opposite

2 \to -2, \quad 5 \to -5

Same

x

, opposite

y

is the rule

(x, y) \to (x, -y)

That rule flips points over the

x

-axis

Answer: A reflection across the $x$ -axis

Example 3: is it a rigid motion?

A triangle with sides $3$ , $4$ , and $5$ is dilated by a scale factor of $2$ . Is the dilation a rigid motion?

Multiply each side by the scale factor

6, \; 8, \; 10

The image's sides are longer than the preimage's

6 \neq 3

The figures are similar but not congruent, so the motion is not rigid

Answer: No — a dilation changes size, so it is not a rigid motion

Try one yourself

Common questions

What does the prime symbol in

A'

mean?

It marks the image of point $A$ after a transformation. The preimage keeps plain letters ( $A$ , $B$ , $C$ ) and the image gets primed letters ( $A'$ , $B'$ , $C'$ ), so you always know which vertex came from which.

Which transformations are rigid motions?

Translations, reflections, and rotations. Each one produces an image congruent to the preimage — same side lengths, same angle measures. A dilation is the one transformation in this lesson that is not rigid, because it changes the figure's size.

Does a translation ever change a figure's size or shape?

No. A translation only changes position — every point slides the same distance in the same direction. The image is always congruent to the preimage, no matter how far the figure slides.

How do I tell a reflection from a rotation on a graph?

Check the orientation. A reflection produces a mirror copy — reading the vertices $A$ , $B$ , $C$ around the figure switches direction, from clockwise to counterclockwise or back. A rotation turns the figure but keeps the vertices reading in the same direction around it.

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Introduction to Transformations

The four transformations

Rigid motions

Reading a transformation from coordinates

Worked examples

Example 1: translate a point

Example 2: name the transformation from a rule

Example 3: is it a rigid motion?

Try one yourself

Common questions

Keep learning

Naming & Measuring Angles; the Angle Addition Postulate

Two-Dimensional Figures

Three-Dimensional Figures

Translations

Reflections